1066 



THE IRRIGATION AGE. 



THE PRIMER OF HYDRAULICS* 



By FREDERICK A. SMITH, C. E. 



Article IX. General Hydraulic Principles. 

 1. The Cause of Flow in Fluids and Definitions. 

 The flow of liquids is generally produced by the force 

 of gravity. In Fig. T2 let the vessel AGKL be filled with 

 water to line AG and let EJ be a pipe of considerable length, 

 having tubes C, D and B in connection therewith, and tube 

 EJ is closed at /. Then the water is at rest and stands level 

 in the reservoir and in the tubes. If now the pipe is opened 



at / the water 

 rushes out at / 

 with a certain 

 velocity ; the 

 water drops in 

 the tubes C, D 

 and B to pointr. 

 C, D, B, and if 

 J a line is drawn 

 through these 



three points to 



K 



Fig. 72. 



the vessel AG it strikes at point I 7 , a certain distance lower 

 than the height of the water GL. The height GF is called 

 the velocity head and is that part of the total potential energy 

 of the height GE which is used to produce the velocity in 

 the pipe; the remainder FE is called the pressure head, which 

 is used in overcoming the friction throughout the pipe EJ ; 

 the height of the water in the tubes C, D and B show the 

 decrease in the pressure heads successively; if the line GH 

 is drawn parallel to FB, it will be seen that the velocity head 

 FG is uniform throughout the flow length. 



If the velocity is measured carefully at / it is found that 

 the height GF is in excess of the height of fall required to 

 produce such velocity. If v is the velocity at / then in the 

 the equation : i? = 2gh 



h can be found by dividing both sides by 2g : h=v*/2g; 

 then suppose the velocity T = 4 ft., then h = 16 -=- 64 = .25 ft. 

 or 3 inches. The distance GF would be then 3 inches if there 

 was no further loss in energy ; there occurs, however, an 

 additional loss at point E, which loss is termed the entry-head, 

 and which depends on the form of entry and which may 

 vary from one-half to one-tenth of the velocity head. 



The line FB is called the hydraulic grade line. 



2. Other Hydraulic Terms and Definitions. 



When water flows through a channel conduit or pipe the 



B 



D C 



Fig. 74. 



velocity of the flow depends on several factors namely, the 

 form of channels, the depth of flow, the roughness of the 

 sides of the channel and the grade of the flow line, commonly 

 called the sine of the slope. Thus if the cross-section of 

 a channel is circular, as in Fig. 73, and if the water stands 

 up to the line AB, then the wetted surface along the circum- 

 ference from A to B via C is called the ivetted perimeter; 

 the flow area is the area of the cross-section of the flowing 

 water and if the flow area is divided by the wetted perimeter 

 the quotient gives the hydraulic radius, which will be termed r. 

 Thus, if a pipe of the diameter d is flowing full the wetted 

 perimeter is <{; the area will be 7rrfV4; hence the hydraulic 

 radius r = Trd~/i -f- -n-d =d/4. 



This proves that the hydraulic radius for all cylindrical 

 conduits flowing either full or half full equals one-fourth of 

 the diameter. 



In Fig. 74 is shown a rectangular channel in cross-sec- 



*Copyrights by D. H. Anderson. 



tion ABCD; if the height of the flow reaches to EF, then 

 the hydraulic radius will be found as follows : flow area 

 equals DC X ED and wetted perimeter equals ED + DC + FC, 

 hence : 



r = DC X ED + (ED + DC 

 + FC): 



Should the box flow full, 

 however, so that the water touches 

 the upper surface AB, then the 

 wetted perimeter increases greatly 

 causing a decided drop in the hy- 

 draulic radius. 



For instance, let the box be 2 ft. wide and 3 ft. high then 

 r = 3 X 2 -=- (3 + 2 + 3 + 2 ) = 6 -^ 10 = .6. 

 On the other hand, let the water surface EF be 2 ft. high; 

 then : 



r = 2X2-r-(2 + 2 + 2)=4 : .07. 

 In Fig. 75 is shown a cross-sec- 

 tion of a triangular channel. If the A B 



water stands up to line DE then the t . 



hydraulic radius r is equal to the 

 area of the triangle DEC divided by 

 the wetted perimeter DC + EC. 

 Should the box, however, run full 

 the wetted perimeter is increased by 

 the length of the cover and results 

 in a greatly decreased hydraulic 

 radius. 



3. General Principles. 

 The well-known formula of Kut- 

 ter: 



f = C\/rs 



is at the present time without doubt the most generally used 

 and gives correct results in practice if applied rightly. The 

 various elements entering into its composition must, however, 

 be correctly determined and intelligently handled. In the 

 above equation i' is the mean velocity of the liquid in feet 

 per second, r is the actual hydraulic radius and j is the sine 

 of slope. The coefficient C is a complex quantity and is 

 derived by calculation from the following formula : 



n 41 e + 1 - 811 4. - 00281 

 c = 



Fig. 77. 



In this formula the quantity n is 

 called the coefficient of roughness, j is 

 the sine of the slope and r the hy- 

 draulic radius as explained above. 



The factor n, then, is really the 



most difficult quantity to determine, as its correct selection is 

 very important. There is, however, sufficient experience to 

 be drawn from, so that the degree of roughness or factor n 

 can be selected with considerable precision. 



In this book the coefficient C has been calculated for 

 factors of roughness varying from .009 to .050, which 

 embraces all imaginable practical channels. Since the varia- 

 tion of the factor C is slight for very considerable differences 

 in slopes, the factor C has been determined only 'for six dif- 

 ferent slopes which covers the field fully as any intermediate 

 slopes can be easily interpolated when necessary. This will 

 later be shown by practical problems. 



The tables have been computed for the square root of r 

 instead of r, as this gives a considerable wider range of appli- 

 cation, and the factor C is found with sufficient accuracy by 

 interpolation between the values shown. 



4. Forms of Of en Channels. 



The form of a channel is an important factor for the 

 Mow of water, and we will consider the characteristics of 

 some of the most important ones. All other things considered 

 equal, the velocity of flow grows as the hydraulic radius r 

 grows, which_is plainly seen from the fundamental formula : 



hence the larger r the better the flow conditions will be. To 

 compare the relative efficiency of various forms we will take the 

 triangular, rectangular, trapezoidal and circular sections which 

 are in general use. Let Fig. 76 show the cross-section of a 

 triangular channel, the sides of which incline 45 to the 

 horizon and form an angle of 90 with each other. This 

 kind of section is best adapted when a considerable velocity 

 of flow is required with a small volume of fluid. To prove 

 this it is easily seen that the factor r grows or falls in a 

 constant ratio, no matter how high the water rises or how 



